发表机构
Myong Ji University; Chung-Ang University（明知大学; 中央大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究维数≥4且具非负截面曲率的紧致真空静态空间，证明正截面曲率且Weyl曲率张量完全散度非负的这类空间等距于标准球面，还推广了临界点方程相关的刚性结果。
AI 中文摘要
本文研究维数≥4且具有非负截面曲率的紧致真空静态空间。首先，证明若n维紧致真空静态空间具有正截面曲率且其Weyl曲率张量的完全散度非负，则它等距于标准球面；在稍弱的非负截面曲率条件下，这类真空静态空间必具有平行Ricci曲率张量。其次，考察由限制在单位体积且常标量曲率的黎曼度量空间上的总标量曲率泛函的临界度量导出的临界点方程，证明具有非负截面曲率且Weyl曲率张量完全散度非负、容许临界点方程非平凡解的紧致黎曼流形必为爱因斯坦流形且等距于标准球面，所得结果可视为文献[4]中结果的推广。
英文摘要
In this paper, we study compact vacuum static spaces of dimension $\ge 4$ with nonnegative sectional curvature. First, we prove that an $n$-dimensional compact vacuum static space is isometric to a standard sphere provided it has positive sectional curvature and the complete divergence of its Weyl curvature tensor is nonnegative. Under the slightly weaker condition of nonnegative sectional curvature, we show that such a vacuum static space must have parallel Ricci curvature tensor. Second, we examine the critical point equation arising from the critical metrics of the total scalar curvature functional on the space of Riemannian metrics restricted to unit volume and constant scalar curvature. We demonstrate that a compact Riemannian manifold with nonnegative sectional curvature and nonnegative complete divergence of the Weyl curvature tensor, which admits a nontrivial solution to the critical point equation, must be Einstein and is isometric to a standard sphere. Our results can be viewed as extensions of the results in [4].
